fractal_core_shell

Scattering from a fractal structure formed from core shell spheres

Parameter

Description

Units

Default value

scale

Scale factor or Volume fraction

None

1

background

Source background

cm-1

0.001

radius

Sphere core radius

Å

60

thickness

Sphere shell thickness

Å

10

sld_core

Sphere core scattering length density

10-6Å-2

1

sld_shell

Sphere shell scattering length density

10-6Å-2

2

sld_solvent

Solvent scattering length density

10-6Å-2

3

volfraction

Volume fraction of building block spheres

None

0.05

fractal_dim

Fractal dimension

None

2

cor_length

Correlation length of fractal-like aggregates

Å

100

The returned value is scaled to units of cm-1 sr-1, absolute scale.

Definition Calculates the scattering from a fractal structure with a primary building block of core-shell spheres, as opposed to just homogeneous spheres in the fractal model. It is an extension of the well known Teixeira[1] fractal model replacing the \(P(q)\) of a solid sphere with that of a core-shell sphere. This model could find use for aggregates of coated particles, or aggregates of vesicles for example.

\[I(q) = P(q)S(q) + \text{background}\]

Where \(P(q)\) is the core-shell form factor and \(S(q)\) is the Teixeira[1] fractal structure factor both of which are given again below:

\[\begin{split}P(q) &= \frac{\phi}{V_s}\left[3V_c(\rho_c-\rho_s) \frac{\sin(qr_c)-qr_c\cos(qr_c)}{(qr_c)^3}+ 3V_s(\rho_s-\rho_{solv}) \frac{\sin(qr_s)-qr_s\cos(qr_s)}{(qr_s)^3}\right]^2 \\ S(q) &= 1 + \frac{D_f\ \Gamma\!(D_f-1)}{[1+1/(q\xi)^2]^{(D_f-1)/2}} \frac{\sin[(D_f-1)\tan^{-1}(q\xi)]}{(qr_s)^{D_f}}\end{split}\]

where \(\phi\) is the volume fraction of particles, \(V_s\) is the volume of the whole particle, \(V_c\) is the volume of the core, \(\rho_c\), \(\rho_s\), and \(\rho_{solv}\) are the scattering length densities of the core, shell, and solvent respectively, \(r_c\) and \(r_s\) are the radius of the core and the radius of the whole particle respectively, \(D_f\) is the fractal dimension, and \(\xi\) the correlation length.

Polydispersity of radius and thickness are also provided for.

This model does not allow for anisotropy and thus the 2D scattering intensity is calculated in the same way as 1D, where the \(q\) vector is defined as

\[q = \sqrt{q_x^2 + q_y^2}\]

Our model is derived from the form factor calculations implemented in IGOR macros by the NIST Center for Neutron Research[2]

../../_images/fractal_core_shell_autogenfig.png

Fig. 103 1D plot corresponding to the default parameters of the model.

Source

fractal_core_shell.py \(\ \star\ \) fractal_core_shell.c \(\ \star\ \) fractal_sq.c \(\ \star\ \) core_shell.c \(\ \star\ \) sas_gamma.c \(\ \star\ \) sas_3j1x_x.c

References

Authorship and Verification

  • Author: NIST IGOR/DANSE Date: pre 2010

  • Last Modified by: Paul Butler and Paul Kienzle Date: November 27, 2016

  • Last Reviewed by: Paul Butler and Paul Kienzle Date: November 27, 2016